As is well known, the theory of partial differential equations is much more difficult than the theory of ordinary differential equations, because PDEs don't behave as nicely.
I was thinking, however, that the generalization of regular derivatives to the higher-dimensional case is usually best thought of as the total (i.e. Frechet) derivative, rather than partial derivatives.
Question: Is there a theory of total differential equations? If not, why not? And if so, does the theory allow one to sidestep around a lot of the problems which arise in the theory of PDEs?
The main problem I could think of is "type-checking"; namely the original function will be a first-order tensor field, the first derivative would be a second-order tensor field, the second derivative would be a third order tensor field, and so on... Is there not a way to circumvent this problem?
Note: Just so everyone is clear, I am not arguing against the usefulness of the theory of PDEs; on the contrary, they were even used by Perelman to prove the Poincare Conjecture. I'm just wondering, if total differential equations would be easier to study, why no one has studied them.
The existence of partial derivatives is only necessary but not sufficient for the existence of the total derivative. So my conjecture is that solving PDE's restricted to the solution space of total differential equations would be much easier and would have to account for far fewer pathological cases, since the existence of the total derivative implies far more regularity properties than one gets by assuming the existence of the partial derivatives alone. (In fact, the existence of partial derivatives does not even imply the existence of all directional derivatives, which in turn does not even imply Gateaux differentiability, which in turn does not even imply total differentiability, so one should expect the general theory of PDEs to have far more pathological cases than the specific theory of total differential equations.)
In other words, it seems like studying total differential equations might make it easier to identify which classes of PDE are easy to solve and would help explain in part why some PDEs are more difficult to solve, since it would make much more explicitly clear how analogies with ODEs break down in those cases. So even though total differential equations would arguably be a sub-case of the field of PDEs, the additional regularity properties would still make it a worthwhile object of study, the same way that the study of symmetric or diagonalizable matrices illuminates the whole of linear algebra.